As mentioned in the previous post, I’ve been working on designing a microscope to be built on an optical rail. As part of the design, I’ve needed to calculate a bunch of distances and sizes – for instance, the size of the back focal plane – that are not usually provided by the objective manufacturer, but that are easy to calculate. So that you won’t have to hunt down all the necessary formulas (most are in chapter 9 of the Handbook of Biological Confocal Microscopy), I thought I would reproduce them here.
The diagram above shows a schematic of an infinity corrected microscope, consisting of an objective and tube lens. Here are some of the basic parameters of the objective, and how they related to each other and to the ray diagram above:
NA = n sin α
NA: Numerical Aperture of the objective
n: refractive index (1 for an air lens)
α: largest angle the objective can collect
M = ft / fo
M: Objective magnification
fo: Objective focal length
ft: Tube lens focal length (200 mm for Nikon)
FOV = fN / M
FOV: Field of View
fN: Field number (25 mm for Nikon)
dBFP = 2 fo sin α
dBFP: Objective back focal plane diameter
β = arctan (FOV / 2fo)
β: Divergence angle of ray bundle
Width of ray bundle at distance d: dBFP + 2d tan(β) = dBFP + d FOV / fo
From this last equation, we can see that putting a stop at some distance from the objective that cuts off part of the ray bundle will result in vignetting of the field of view. This is particularly useful to know when designing your own optical system, so that you don’t accidentally cut off part of the field of view.
With these equations in place, lets calculate them for a Nikon 10x / 0.3 NA objective.
- From the NA, and the fact that this is an air lens (n = 1), we can calculate α = 17.45 degrees.
- From the magnification, and knowing that Nikon uses a 200 mm tube lens, we know that fo = 20 mm
- The field of view is 2.5 mm (from the magnification and the 25 mm Nikon field number)
- The back focal plane diameter is 12 mm
- And β, the beam divergence, is 3.6 degrees.